Generation of Zonal Flows in a Rotating Self-Gravitating Fluid

Abstract

We demonstrate the possibility of generation of zonal (shear) flows in a rotating self-gravitating fluid. A set of equations describing the nonlinear interaction between a large-scale zonal flow (ZF) and a small-scale drift-gravity wave is derived. A nonlinear dispersion relation is obtained, from which the possible instability of the ZF follows. The necessary condition for instability in the space of wave numbers of the drift-gravity wave, as well as the instability threshold for the wave amplitude, are obtained. The growth rate of the modulation instability of ZF is found. The generation of ZFs is due to the Reynolds stresses produced by finite amplitude drift-gravity waves.

1. Introduction

Zonal flows (ZFs) are large-scale flows in the atmospheres of rotating planets with a nonuniform velocity profile and are found mainly in the latitudinal direction (along the meridians) [1,2,3,4,5]. In plasmas, ZFs are azimuthally symmetric modes that depend only on the radial coordinate and play a crucial role in regulating the nonlinear evolution of drift-wave instabilities in the devices with magnetic plasma confinement [6,7,8]. Zonal flows are clearly observed in the atmosphere of Jupiter and Saturn. For the Earth’s atmosphere, these correspond, for example, to the trade winds, the so-called westerly winds and stratospheric polar jets in Antarctica. The experimentally observed transition to the improved plasma confinement regime in tokamaks is associated with the emergence of ZF in the poloidal direction, which suppresses fluctuations and creates a barrier to turbulent transport. For the theoretical description of zonal and vortex motions in a rotating atmosphere, the so-called quasi-geostrophic approximation is often used, when the characteristic frequencies of disturbances are much lower than the rotation frequency. For plasmas, the corresponding rotation frequency is the ion gyrofrequency. The simplest model in this case is the Charney equation [9] (for plasma known as the Hasegawa–Mima equation [10]).

The generation of ZFs is not fully understood. For example, it is known that they can be caused by nonuniform heating of the atmospheric layers by solar radiation [11]. An effective mechanism for generating ZFs due to the modulation instability of drift waves in plasma was studied in Refs. [6,7] (see also numerous references in the review [8]). Padma Kant Shukla and Lennart Stenflo were the first to show [12] that ZFs in the atmosphere can be effectively excited by modulation instability of finite-amplitude Rossby waves. The driving mechanism of this instability is due to the Reynolds stresses [12,13]. Then the generation of ZFs by magnetized Rossby waves was considered for the Earth’s ionospheric E-layer [14,15].

The dynamics of self-gravitating systems were first studied by Sir James Jeans works [16]. Later, in the studies [17,18,19,20,21,22,23], coherent structures in such systems in the form of dipole vortices and vortex chains were investigated. In this paper, we consider the possibility of generating ZFs in such a system and show that under certain conditions and exceeding the amplitude threshold, the generation of ZF is possible due to modulation instability of the drift-gravity wave. Let us point out that the generation of large-scale ZF is a manifestation of the inverse cascade in two-dimensional drift-gravity turbulence.

The paper is organized as follows. In Section 2, we present the system of equation to describe the interaction between ZFs and drift-gravity waves in a rotating self-gravitating fluid. The nonlinear dispersion equation, thresholds, and growth rates of ZF instability are obtained in Section 3. Finally, Section 4 concludes the paper.

2. Model Equations

A system of coupled three-dimensional nonlinear equations for the gravitational potential and the z-component of the fluid velocity was obtained for perturbations in self-gravitating rotating systems [24]. We modified this system by making the following assumptions: the self-gravitating fluid rotates with a constant angular velocity ${\mathrm{\Omega }}_0={\mathrm{\Omega }}_0\widehat{\mathbf{z}}$ in a plane perpendicular to the $\widehat{\mathbf{z}}$ axis; the inhomogeneity of the equilibrium density in the radial direction is weak and is characterized by the length of the inhomogeneity L (all characteristic scales of perturbations are much greater than L); the characteristic frequencies of perturbations are small compared to the rotation frequency. Under these assumptions, we showed in Ref. [25] that the system obtained in Ref. [24] is reduced to the following dimensionless equation for the gravitational potential:

$$\begin{array}{c}{\displaystyle \frac{\partial }{\partial t}}(\alpha \Delta \psi -{\Delta }^2\psi )-v_{*}{\displaystyle \frac{\partial }{\partial y}}(\psi +\Delta \psi )\\ +\{\psi +\Delta \psi ,\alpha \Delta \psi -{\Delta }^2\psi \}=0,\end{array}$$

(1)

where $\mathrm{\Delta }={\partial }^2/\partial x^2+{\partial }^2/\partial y^2$ is the 2D (two-dimensional) Laplacian, $\alpha =4{\mathsf{\Omega }}_0^2/{\omega }_0^2-1$, and $v_{*}={\lambda }_J/L$. Here ${\omega }_0$ is the so-called Jeans frequency defined by ${\omega }_0=2\sqrt{\pi G{\rho }_0}$, where G is the gravitational constant, ${\lambda }_J=c_s/{\omega }_0$ is the Jeans length, $c_s$ is the speed of sound in an adiabatic medium of uniform density ${\rho }_0$. Dimensionless time t, spatial coordinates $\mathbf{r}=(x,y)$, and gravitational potential $\psi$ are introduced by

$$t=2{\mathsf{\Omega }}_0{t}_{\mathbf{1}},\mathbf{r}={\displaystyle \frac{{\mathbf{r}}_{\mathbf{1}}}{{\lambda }_J}},\mathrm{and}\psi ={\displaystyle \frac{{\omega }_0^2}{4{\mathsf{\Omega }}_0^2{c}_s^2}}{\psi }_{\mathbf{1}},$$

(2)

where variables $t_{\mathbf{1}},{\mathbf{r}}_{\mathbf{1}}$, and ${\psi }_{\mathbf{1}}$ correspond to dimensional variables. The nonlinear term in Equation (1) is written in the form of the Poisson bracket (Jacobian) defined as

$$\{f,g\}={\displaystyle \frac{\partial f}{\partial x}}{\displaystyle \frac{\partial g}{\partial y}}-{\displaystyle \frac{\partial f}{\partial y}}{\displaystyle \frac{\partial g}{\partial x}}.$$

(3)

In the linear approximation, taking $\psi \sim \text{exp}(i\mathbf{k}·\mathbf{r}-i\omega t)$, where $\omega$ and $\mathbf{k}=(k_x,k_y)$ are the frequency and wave vector, respectively, Equation (1) gives the dispersion relation for the drift-gravity wave [24]

$${\omega }_{\mathbf{k}}={\displaystyle \frac{k_y{v}_{*}(1-k^2)}{k^2(\alpha +k^2)}}.$$

(4)

Solutions of Equation (1) for a rotating self-gravitating fluid were found in the form of fourth order dipole vortex solitons in Ref. [25]. Here, however, we are interested in the possibility of generating ZFs in such a system.

The key point, alike in Ref. [12], is the separation of scales for drift-gravity waves and ZF. Assuming that ZF varies on a much larger time scale than the drift-gravity wave, one can decompose the gravity potential $\psi$ into “slow” and “fast” parts, that is $\psi =\widehat{\psi }+\tilde{\psi }$, where $\widehat{\psi }$ corresponds to the large-scale ZF and $\tilde{\psi }$ to the short-scale drift-gravity wave. Averaging Equation (1) over the short scales, one obtains the evolution equation for the ZFs: revised

$${\displaystyle \frac{\partial }{\partial t}}(\alpha \Delta \widehat{\psi }-{\Delta }^2\widehat{\psi })=-\{\tilde{\psi }+\Delta \tilde{\psi },\alpha \Delta \tilde{\psi }-{\Delta }^2\tilde{\psi }\}.$$

(5)

The term in the right-hand side of Equation (5) represents the averaged Reynolds stresses of rapidly oscillating drift-gravity waves. In turn, the dynamic equation for $\tilde{\psi }$ taking into account the nonlinear interaction between the ZF and the drift-gravity wave is

$$\begin{array}{c}{\displaystyle \frac{\partial }{\partial t}}(\alpha \Delta \tilde{\psi }-{\Delta }^2\tilde{\psi })-v_{*}{\displaystyle \frac{\partial }{\partial y}}(\psi +\Delta \tilde{\psi })\\ +\{\widehat{\psi }+\Delta \widehat{\psi },\alpha \Delta \tilde{\psi }-{\Delta }^2\tilde{\psi }\}\\ +\{\tilde{\psi }+\Delta \tilde{\psi },\alpha \Delta \widehat{\psi }-{\Delta }^2\widehat{\psi }\}=0.\end{array}$$

(6)

The system of Equations (5) and (6) is a closed nonlinear system of equations describing the dynamics of large-scale and small-scale motions within the framework of the model (1). In Section 3 just below, we consider the interaction of monochromatic waves.

3. Nonlinear Dispersion Relation and Instability of Zonal Flows

The amplitude of the zonal flow is represented as

$$\widehat{\psi }={\psi }_{\mathbf{q}}\text{exp}(i\mathbf{q}·\mathbf{r}-i\mathsf{\Omega }t)+{\psi }_{\mathbf{q}}^{*}\text{exp}(-i\mathbf{q}·\mathbf{r}+i\mathsf{\Omega }t),$$

(7)

where $\mathbf{q}=q\widehat{\mathbf{x}}$ and $\widehat{\mathbf{x}}$ is the unit vector in the x-direction. The drift-gravity wave is considered as a superposition of the pump wave and two sidebands, that is $\tilde{\psi }={\psi }_0+{\psi }_{+}+{\psi }_{-}$. For the pump wave and sidebands, one has

$${\psi }_0={\psi }_{\mathbf{k}}\text{exp}(i\mathbf{k}·\mathbf{r}-i{\omega }_{\mathbf{k}}t)+{\psi }_{\mathbf{k}}^{*}\text{exp}(-i\mathbf{k}·\mathbf{r}+i{\omega }_{\mathbf{k}}t)$$

(8)

and

$${\psi }_{\pm }={\psi }_{\mathbf{k}\pm }\text{exp}(i{\mathbf{k}}_{\pm }·\mathbf{r}-i{\omega }_{\mathbf{k}\pm }t)+{\psi }_{\mathbf{k}\pm }^{*}\text{exp}(-i{\mathbf{k}}_{\pm }·\mathbf{r}+i{\omega }_{\mathbf{k}\pm }t),$$

(9)

respectively, where ${\mathbf{k}}_{\pm }=\mathbf{k}\pm \mathbf{q}$ and ${\omega }_{\mathbf{k}\pm }={\omega }_{\mathbf{k}}\pm \mathsf{\Omega }$. Substituting Equations (7)–(9) into Equation (5), one finds for the zonal flow amplitude:

$$\begin{array}{c}\mathsf{\Omega }{\psi }_{\mathbf{q}}=-{\displaystyle \frac{i{[\mathbf{k}\times \mathbf{q}]}_z}{q^2(\alpha +q^2)}}\\ \times \{[(1-k^2)(\alpha +k_{+}^2)k_{+}^2-(1-k_{+}^2)(\alpha +k^2)k^2]{\psi }_0^{*}{\psi }_{+}\\ -[(1-k^2)(\alpha +k_{-}^2)k_{-}^2-(1-k_{-}^2)(\alpha +k^2)k^2]{\psi }_0{\psi }_{-}^{*}\}.\end{array}$$

(10)

The Fourier amplitudes ${\psi }_{\mathbf{k}+}$ and ${\psi }_{\mathbf{k}-}^{*}$ are found from Equation (6):

$$\begin{array}{c}{\psi }_{\mathbf{k}+}=-i{\displaystyle \frac{{[\mathbf{k}\times \mathbf{q}]}_z(k^2-q^2)(\alpha +k^2+q^2-k^2{q}^2)}{(\mathsf{\Omega }+\delta {\omega }_{+})k_{+}^2(\alpha +k_{+}^2)}}{\psi }_{\mathbf{k}}{\psi }_{\mathbf{q}},\end{array}$$

(11)

$$\begin{array}{c}{\psi }_{\mathbf{k}-}^{*}=i{\displaystyle \frac{{[\mathbf{k}\times \mathbf{q}]}_z(k^2-q^2)(\alpha +k^2+q^2-k^2{q}^2)}{(\mathsf{\Omega }-\delta {\omega }_{-})k_{-}^2(\alpha +k_{-}^2)}}{\psi }_{\mathbf{k}}^{*}{\psi }_{\mathbf{q}},\end{array}$$

(12)

where $\delta {\omega }_{\pm }={\omega }_{\mathbf{k}}-{\omega }_{\mathbf{k}\pm \mathbf{q}}$. Substituting Equations (11) and (12) into Equation (10), one finds

$$\mathsf{\Omega }=A|{\psi }_0{|}^2\left({\displaystyle \frac{B_{\mathbf{k}+\mathbf{q}}}{\mathsf{\Omega }+{\omega }_{\mathbf{k}}-{\omega }_{\mathbf{k}+\mathbf{q}}}}+{\displaystyle \frac{B_{\mathbf{k}-\mathbf{q}}}{\mathsf{\Omega }-{\omega }_{\mathbf{k}}+{\omega }_{\mathbf{k}-\mathbf{q}}}}\right),$$

(13)

where

$$\begin{array}{c}A={\displaystyle \frac{{[\mathbf{k}\times \mathbf{q}]}^2(q^2-k^2)(\alpha +k^2+q^2-k^2{q}^2)}{q^2(\alpha +q^2)}},\\ B_{\mathbf{k}\pm \mathbf{q}}=1-k^2-{\displaystyle \frac{(1-k_{\pm }^2)(\alpha +k^2)k^2}{(\alpha +k_{\pm }^2)k_{\pm }^2}}\end{array}$$

(14)

$$\begin{array}{c}={\displaystyle \frac{(1-k^2)({\omega }_{\mathbf{k}}-{\omega }_{\mathbf{k}\pm \mathbf{q}})}{{\omega }_{\mathbf{k}}}}.\end{array}$$

(15)

Combining Equations (13) and (15) gives

$$1={\displaystyle \frac{A(1-k^2)(\delta {\omega }_{+}+\delta {\omega }_{-}){\left|{\psi }_0\right|}^2}{{\omega }_{\mathbf{k}}(\mathsf{\Omega }+\delta {\omega }_{+})(\mathsf{\Omega }-\delta {\omega }_{-})}}.$$

(16)

Equation (16) is a quadratic equation in $\mathsf{\Omega }$ and it can be straightforwardly solved. The negativity of the discriminant of the equation corresponds to instability with the growth rate $\gamma =\mathrm{Im}\mathsf{\Omega }$. However, of greatest interest is the case $q\ll k$, which corresponds to a large-scale zonal flow. In this case, using

$${\omega }_{\mathbf{k}\pm \mathbf{q}}\sim {\omega }_{\mathbf{k}}\pm {\displaystyle \frac{\partial {\omega }_{\mathbf{k}}}{\partial \mathbf{k}}}·\mathbf{q}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial }^2{\omega }_{\mathbf{k}}}{\partial {\mathbf{k}}^2}}{q}^2,$$

(17)

one obtains from Equations (14) and (16):

$$\begin{array}{c}{(\mathsf{\Omega }-q{v}_g)}^2-{\displaystyle \frac{{\left(v_g^{\prime }\right)}^2}{4}}{q}^4\\ ={\displaystyle \frac{{[\mathbf{k}\times \mathbf{q}]}^2{v}_g^{\prime }{k}^2(1-k^2)[\alpha +k^2(1-q^2)]{\left|{\psi }_0\right|}^2}{{\omega }_{\mathbf{k}}(\alpha +q^2)}},\end{array}$$

(18)

where $v_g=\partial {\omega }_{\mathbf{k}}/\partial k_x$ is the group velocity of the drift-gravity wave in the x direction, and $v_g^{\prime }={\partial }^2{\omega }_{\mathbf{k}}/\partial k_x^2$. Explicit expressions for $v_g$ and $v_g^{\prime }$ are given by

$$\begin{array}{c}{v}_g={\displaystyle \frac{2k_x{k}_y{v}_{*}[k^2(k^2-2)-\alpha ]}{k^4{(\alpha +k^2)}^2}},\\ v_g^{\prime }={\displaystyle \frac{2k_y{v}_{*}}{k^6{(\alpha +k^2)}^3}}[k^2(k^2+\alpha )(k^4+12k_x^2)+4{\alpha }^2{k}_x^2\end{array}$$

(19)

$$\begin{array}{c}-2k^6(1+2k_x^2)-\alpha k^2(\alpha +3k^2)].\end{array}$$

(20)

Thus, it is seen from Equation (18) that instability of the zonal flows (ZF) occurs provided

$${\displaystyle \frac{{\left(v_g^{\prime }\right)}^2}{4}}{q}^4+{\displaystyle \frac{{[\mathbf{k}\times \mathbf{q}]}^2{v}_g^{\prime }{k}^4(\alpha +k^2)[\alpha +k^2(1-q^2)]{\left|{\psi }_0\right|}^2}{k_y{v}_{*}(\alpha +q^2)}}<0.$$

(21)

Next, the following remark should be made regarding the parameter $\alpha$. For example, for typical parameters of spiral galaxies, one has ${\rho }_0\approx 5·{10}^{-22}\mathrm{g}/{\mathrm{cm}}^3$ and $c_s\approx 5·{10}^6\mathrm{cm}/\mathrm{s}$; then, the Jeans frequency is estimated as ${\omega }_0\approx 2·{10}^{-14}{\mathrm{s}}^{-1}$ with ${\lambda }_J\approx 2·{10}^{20}\mathrm{cm}$. So, the typical rotation frequency ${\mathsf{\Omega }}_0$ is of the same order of magnitude as the Jeans frequency ${\omega }_0$. Under such conditions, the parameter $\alpha$ can be of the order of or much greater than unity and have different signs. The instability picture strongly depends on the sign of the parameter $\alpha$ and, as can be seen, becomes more complicated for $\alpha <0$. Note that the applicability of the model is violated when $\alpha +k^2\sim 0$ or $\alpha +q^2\sim 0$. In this paper, we restrict ourselves to the case $\alpha >1$.

Large-scale ZF implies in most cases $q\ll 1$; that is, the characteristic size of the zonal structure exceeds the Jeans length ${\lambda }_J$. In this case, it follows from Equation (21) that instability is possible only if $v_g^{\prime }<0$ and when the amplitude of the drift-gravity wave exceeds the threshold stability,

$${\displaystyle \frac{{[\mathbf{k}\times \mathbf{q}]}^2\left|v_g^{\prime }\right|k^4{(\alpha +k^2)}^2}{k_y{v}_{*}(\alpha +q^2)}}{\left|{\psi }_0\right|}^2>{\displaystyle \frac{{\left(v_g^{\prime }\right)}^2}{4}}{q}^4.$$

(22)

As follows from Equation (20), the sign of $v_g^{\prime }$ is determined by the function

$$\begin{array}{c}Q(k_x,k_y)=k^2(k^2+\alpha )(k^4+12k_x^2)+4{\alpha }^2{k}_x^2\\ -2k^6(1+2k_x^2)-\alpha k^2(\alpha +3k^2).\end{array}$$

(23)

A necessary but insufficient condition for instability corresponds to $Q<0$. The curves $Q(k_x,k_y)=0$ on the plane $(k_x,k_y)$ for $\alpha =5$ and $\alpha =20$ are shown in Figure 1. The behavior of the function Q from the wave numbers of the drift-gravity wave $k_x$ and $k_y$ does not change qualitatively in a sufficiently large range of variation of the parameter $\alpha$. If $\alpha \gg 1$ and $k_x,k_y\approx 1$, one obtains a simplified expression $Q(k_x,k_y)=3k_x^2-k_y^2$. Interestingly, in this limit the condition $Q<0$ coincides with the necessary condition for instability of the zonal Rossby flow [13]. Taking into account Equation (22), one has from Equation (18)

$${\mathsf{\Omega }}_{\pm }=q{v}_g\pm i{\left\{{\displaystyle \frac{{[\mathbf{k}\times \mathbf{q}]}^2|v_g^{\prime }|k^4{(\alpha +k^2)}^2{\left|{\psi }_0\right|}^2}{k_y{v}_{*}(\alpha +q^2)}}-{\displaystyle \frac{{\left(v_g^{\prime }\right)}^2}{4}}{q}^4\right\}}^{1/2},$$

(24)

where instability corresponds to the upper sign in Equation (24). The instability growth rate is given by

$$\gamma ={\left[{\displaystyle \frac{q^2{k}_y|v_g^{\prime }|k^4{(\alpha +k^2)}^2{\left|{\psi }_0\right|}^2}{v_{*}\alpha }}-{\displaystyle \frac{{\left(v_g^{\prime }\right)}^2}{4}}{q}^4\right]}^{1/2},$$

(25)

where we take into account that $q\ll \alpha$ since $\alpha >1$. The dependence of the instability growth rate $\gamma$ on the wave number of perturbations q in the case $\alpha =5$ for different values $|{\psi }_0{|}^2$ and for specific values $k_x=0.2$ and $k_y=1$ is shown in Figure 2. The optimal wave number $q_{\mathrm{opt}}$ corresponding to the maximum growth rate is found from Equation (25):

$$q_{\mathrm{opt}}={\left[{\displaystyle \frac{2k_y|v_g^{\prime }|k^4{(\alpha +k^2)}^2{\left|{\psi }_0\right|}^2}{v_{*}\alpha {\left(v_g^{\prime }\right)}^2}}\right]}^{1/2},$$

(26)

and then substituting Equation (26) into Equation (25) gives the maximum instability growth rate

$${\gamma }_{\max }={\displaystyle \frac{k_y{k}^4{(\alpha +k^2)}^2{\left|{\psi }_0\right|}^2}{v_{*}\alpha }}.$$

(27)

The rapid power-law increase in the instability growth rate of the ZF with decreasing scales of the drift-gravity waves signifies, similar to that for Rossby waves in Refs. [12,13], the manifestation of the inverse cascade, i.e., the transfer of spectral energy of short-wave drift-gravity motions (turbulence) into long-scale ZFs.

4. Conclusions

In this paper, have demonstrated the possibility of generation of ZFs (shear flows) in a rotating self-gravitating fluid when the corresponding rotation frequency is much greater than the characteristic frequencies of disturbances (analogous to the quasi-geostrophic approximation in geophysics). A set of equations describing the nonlinear interaction between a large-scale ZF and a small-scale drift-gravity wave has been derived. Accordingly, we have considered the interaction of a pump drift-gravity wave, two satellites of the pump waves (side-band waves) and ZF. A nonlinear dispersion relation has been obtained, from which the possible instability of the ZF follows. The necessary condition for instability in the space of wave numbers of the drift-gravity wave, as well as the instability threshold for the wave amplitude, has been obtained. The growth rate of modulation instability of ZF has been found. The generation of ZFs is due to the Reynolds stresses produced by finite amplitude drift-gravity waves. The consequence of the investigated instability is the emergence of the inverse cascade of energy; that is, the generation of large-scale structures in the form of ZFs from small-scale turbulence.

Note that we have considered the case when for the dimensionless parameter $\alpha$, which determines the relationship between the rotation frequency of the system and the Jeans frequency, the condition $\alpha >1$ holds. In reality, the opposite case $\alpha <0$ is also possible. The corresponding analysis of threshold conditions and finding the growth rate is considered to be performed elsewhere.